Smooth

Definition

A smooth function is one that has continuous derivatives up to the required order over a given interval. In calculus, this typically means the function is differentiable and its derivative is also continuous.

5 Must Know Facts For Your Next Test

1. For a curve to be considered smooth, it must have no sharp corners or cusps within the interval of consideration.
2. A smooth function used in arc length calculations ensures that the integral for arc length converges properly.
3. Smoothness of a curve implies the existence of higher-order derivatives which are necessary for accurate surface area computations.
4. The concept of a smooth function can be extended to multivariable functions when computing surface area by ensuring partial derivatives are continuous.
5. In applications of integration, smoothness helps in simplifying the use of parametrization techniques for curves and surfaces.

Review Questions

• Why is it essential for a curve to be smooth when calculating its arc length?
• How does the smoothness of a function affect the computation of surface areas in calculus?
• What role do higher-order derivatives play in determining if a curve or surface is smooth?

Related terms

Differentiable: A function is differentiable at a point if it has a defined derivative at that point.

Continuous Function: A function is continuous if there are no breaks, jumps, or holes in its graph.

$C^n$ Class Functions: $C^n$ class functions have continuous derivatives up to the $n^{th}$ order.